Abstract
Most of modern technological networks that can perform their tasks with various distinctive levels of efficiency are multistate networks, and reliability is a fundamental attribute for their safe operation and optimal improvement. For a multistate network, the two-terminal reliability at demand level d, defined as the probability that the network capacity is greater than or equal to a demand of d units, can be calculated in terms of multistate minimal paths, called d-minimal paths (d-MPs) for short. This paper presents an efficient algorithm to find all d-MPs for the multistate two-terminal reliability problem. To advance the solution efficiency of d-MPs, an improved model is developed by redefining capacity constraints of network components and minimal paths (MPs). Furthermore, an effective technique is proposed to remove duplicate d-MPs that are generated multiple times during solution. A simple example is provided to demonstrate the proposed algorithm step by step. In addition, through computational experiments conducted on benchmark networks, it is found that the proposed algorithm is more efficient.
Highlights
Ere are two important mathematical models with respect to the d-minimal paths (d-MPs) problem. e first mathematical model for solving d-MPs is originally proposed by Lin et al [22]
An MP is a subset of edges, such that if any edge is removed from it, the remaining is no longer a path. e second mathematical model for solving d-MPs is proposed by Yeh [23]. is model without requiring MPs information is a componentbased formulation with three constraints
Through computational experiments, it is found that the proposed algorithm is more efficient in finding all d-MPs
Summary
E multistate network discussed satisfies the following assumptions [28, 30]. (1) Each node is perfectly reliable, which means no capacity constraint is imposed on nodes. (2) e state/capacity of each edge ei(1 ≤ i ≤ m) is a nonnegative integer-valued random variable which takes values from 0 to Wi according to a given distribution. (3) e states/capacities of different edges are statistically independent. (4) All flows in the network obey the conservation law, i.e., total flows into and from a node (not source and sink nodes) are all equal. (4) All flows in the network obey the conservation law, i.e., total flows into and from a node (not source and sink nodes) are all equal. Pp, from the source node s to the sink node t in the network, and the flow travelling through Pj(1 ≤ j ≤ p) is denoted by fj(1 ≤ j ≤ p). Since each unit of flow sent from source node s to sink node t should travel through one of the MPs, the summation of flows travelling through all MPs must be equal to d, which is reflected by condition (3). E direct verification method is mainly based on the definition of d-MPs, that is, every d-MP candidate x derived from Lemma 1 is verified to determine whether it satisfies M(x − 0(ei)) < d for each xi > 0. In contrast to the direct verification method and the comparison method, the cycle-checking method holds the efficiency advantage, but it is more applicable to directed networks. e set of d-MPs derived from Step 2 may contain duplicate d-MPs which significantly contribute to the computational burden of reliability evaluation but have no effect on the final reliability value, and there is need to remove duplicate d-MPs (STEP 3)
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